Problem: Find $\lim_{x\to\infty}\dfrac{\cos(x)-x}{x+2}$. Choose 1 answer: Choose 1 answer: (Choice A) A $-1$ (Choice B) B $0$ (Choice C) C $2$ (Choice D) D The limit doesn't exist
Solution: When dealing with limits that include $\cos(x)$, it's important to remember that $\lim_{x\to\infty}\cos(x)$ doesn't exist, as $\cos(x)$ keeps oscillating between $-1$ and $1$ forever. ${2}$ ${4}$ ${6}$ ${8}$ ${\llap{-}4}$ ${\llap{-}6}$ ${\llap{-}8}$ ${2}$ $y$ $x$ $y=\cos(x)$ This doesn't necessarily mean that our limit doesn't exist. Think what happens to $\dfrac{\cos(x)-x}{x+2}$ as $x$ increases towards positive infinity. $\cos(x)$ oscillates between $-1$ and $1$. This can be represented mathematically by the following double inequality: $\dfrac{-1-x}{x+2}\leq\dfrac{\cos(x)-x}{x+2}\leq\dfrac{1-x}{x+2}$ The result is a graph that's always between the graphs of $y=\dfrac{\pm 1-x}{x+2}$ (the dashed lines). ${2}$ ${4}$ ${6}$ ${8}$ ${\llap{-}2}$ ${\llap{-}4}$ ${\llap{-}6}$ ${\llap{-}8}$ ${2}$ ${4}$ ${6}$ ${8}$ ${\llap{-}2}$ ${\llap{-}4}$ ${\llap{-}6}$ ${\llap{-}8}$ $y$ $x$ Since $\lim_{x\to\infty}\dfrac{\pm 1-x}{x+2}=-1$, so must our limit be equal to $-1$. In conclusion, $\lim_{x\to\infty}\dfrac{\cos(x)-x}{x+2}=-1$.